In Part II of this work we have dealt with the theory of open quantum systems with the particular emphasis on the memory effects. In Chapter 1 we have reviewed the well-known density matrix approach to the theory of open quan-tum systems, we have introduced the RDO and obtained the time non-local QME. We have also introduce the Markov approximation removing this time non-locality.
The central topic of the Chapter 2 was to study the memory effects in the photo-induced vibration dynamics of molecular systems. After discussing the necessary conditions for the memory effects to appear we introduced the molecular system in section 2.1 consisting of two electronic levels modulated by a single vibrational coordinate. With this model in mind we have shortly discussed the memory effects in frequency domain by deriving the cw absorp-tion coefficient in secabsorp-tion 2.2. Starting from the section 2.3 we concentrated exclusively on the time-domain memory effects.
In section 2.3 we derived a more convenient form of the QME getting ex-plicitely rid of the initial correlation term. It was also argued that the non-Markovian dynamics after the ultra-fast optical preparation would be similar to the “artificial” dynamics which appears when neglecting the initial system-bath correlation term in the equation of motion involving memory. This is best seen while considering a δ-pulse. In order to see what kind of dynamics we could expect we solve a special case of an infinitely long memory time and excitation byδ-pulse analytically in section2.4. We have observed an irregular wavepacket motion which can be described as a phase insensitive superposi-tion of two independent wavepackets moving on effective potentials. More im-portantly, the wavepacket motion results in the oscillations of the vibrational populations which exhibit the frequency equal to the half of the vibrational frequency of the system.
The interplay of the memory effects and the optical state preparation by ultra-fast laser pulses has been studied in section2.5. By means of the compar-ison of the results obtained from the full non-Markovian QME with the results following from the Markov approximation we have identified three main mem-ory effects in the vibrational dynamics
• the different layout of the vibrational levels populations in times just after the excitation
• slight change in the life times of the vibrational levels
• change of the amplitude and the period of the oscillations modulating
the overall course of the vibrational level populations
Since in case of an experiment we do not have a comparison with Markovian dynamics only the change of the oscillation period can be taken a character-istics enabling to identify the memory effects independently. The irregular oscillation in the times just after the laser pulse excitation corresponds exactly to that one obtained from the analytical solution in section 2.4. The oscilla-tion period tends to that one of the Markovian dynamics after the time roughly corresponding to the memory time.
Finally, the section2.6 dealt with the extension of the QME to the case of the non-linear system-bath coupling. It was shown that the chosen form of the non-linearity leads to a constant part of the correlation function and thus to the survival of the memory effects to long times. The calculation suggested that in case of the strong non-linear system-bath coupling non-Markovian effects may become predominant.
The memory effects found in Chapter2are the most pronounced for the case of the impulsive excitation. For this case an alternative description using the time-dependent Redfield tensor has been found reproducing non-Markovian re-sults quite well. For the finite laser pulses the correspondence of the rere-sults has been worse, but the memory effects become less pronounced, on the other hand.
Thus, one may conclude, that unless there have to be large non-Markovian ef-fects expected or the computational effort to solve non-Markovian equations is reduced the practical simulations of the photo-induced molecular dynamics may be successfully done within a (special type of) Markovian QME.
The memory effects identified in the dynamics of the studied system can be related to the effective redefinition of the initial state by the action of an ultrafast laser field. The effects appear as the reaction of the system on this sudden change which the system “remembers” at least for the memory time tmem. In the regime of the weak system-bath interaction (which is the only valid here because of the usage of the second order perturbation theory with respect to the system-bath coupling in QME) such a sudden change cannot be induced by the dissipative part. Thus, one can conclude that in case of a weak dissipation the only non-Markovian effects possibly observable are those induced by ultra-fast external fields.
In the next part (Part III) of this work the optimal control theory is ap-plied on the dissipative systems within the Markovian approximation. The generalization to include the memory effects is the next logical step in the con-struction of the theory and a corresponding formulation will be also done in Part III. The inclusion of the memory into the optimal control theory becomes necessary whenever the resulting optimal fields show ultrafast features.
External Field Control of Open Quantum Systems
61
Laser Pulse Control of Molecular Dynamics
The suggestion to the control molecular dynamics by means of ultrashort laser pulses dates back to the middle of the eighties. First control schemes have been results of more or less completely theoretical considerations. A comprehensive overview on all attempts discussed so far has been given recently in [RZ00]. The effort invested into the laser pulse control of the molecular dynamics is mainly motivated by a prospect of the control of chemical reactions [RdVRMK00]. One expects to be able to selectively induce distinct channels of a chemical reaction to enhance the yield of special reaction products and to suppress unwanted ones. This goal should be achieved by means of specially tailored laser pulses.
The idea is usually nicely depicted by a simple ABC model (Fig. 3.1). We start with the molecule consisting of a bond state of three componentsA, B, C and we aim to selectively induce a reaction yielding either a molecule AB and separate fragment C or a molecule AC and a fragmentB.
Originally, several different approaches to the laser pulse control have been suggested, using several different mechanisms to achieve the control goal. In their pioneering work Tannor, Kosloff, and Rice [TKR86] suggested so-called pump-dump scheme. Here, the molecular system is first excited electronically by a short pump pulse. Such a short laser pulse has correspondingly broad frequency spectrum and enables to excite a coherent superposition of several vibrational levels in the excited electronic state. This yields a creation of a wavepacket in a non-equilibrium position on the PES of the excited electronic state and subsequent motion of the wavepacket on a given PES. The system evolves freely on the excited PES and after a particular delaytdit is de-excited using a second (dump) laser pulse. The delay timetdis chosen in such a manner, that the wavepacket has a convenient position in order to make a transition
63
Figure 3.1: Example of the control of chemical reaction. The laser pulse should selectively drive the system into a given reaction chanel.
into a desired reaction chanel. On Figure 3.2 this mechanism is demonstrated for a one-dimensional PES.
A different control scheme this time using continuous lasers has been pro-posed by Brumer and Shapiro. It uses the phase relation between two laser fields to vary the population ratio between two energetically degenerated lev-els [BS86]. The basic idea is that if one finds two independent pathways that connect the same initial and final states of the system, one can modulate the probability of the population of a specific final state. This is possible because the probability of the transition from the initial to the final state is propor-tional to the square of the of the sum of the amplitudes associated with the individual transitions. As an example one can deal with a single- and three–
photon excitation as it is depicted on Fig. (3.3). The control is performed by the laser pulse with frequency 3ω for the resonant excitation and the laser pulse with the frequency ω for three-photon excitation. The target states are denoted |eiand |e0i. The probability of the transition from the initial state to the final state |ei with the energy E can be written as
W(e, E) =W1(e, E) +W3(e, E) +W13(e, E). (3.1) Here, W1(e, E) andW3(e, E) represent the probabilities of the one-photon and three-photon transitions, respectively, andW13(e, E) represents an interference term rising from the simultaneous one-photon and three-photon transitions. In the weak field regime it can be shown [SB92] that W13(e, E) depends on the difference θ3−3θ1, where θ1 and θ3 are the phases of the radiation field of the one-photon and three-photon transitions, respectively. Because the interference term of the transition probabilities to the individual states depends on the mutual phase of the laser fields the ratio
R(e, e0) = W(e, E)
W(e0, E) (3.2)
Q
U(Q)
pumppulsedump pulse delay td
|g>
|e>
Figure 3.2: Example of the control by the pump-dump scheme. To dissociate the molecule one can use two transitions (red arrows) between the ground and the exited state. First, pump pulse creates a wavepacket on the excited PES.
After a delay time td the position of the wavepacket is suitable for the second transition accomplished by the dump pulse. It results in a non-equilibrium position of the wavepacket on the ground state PES leading to the dissociation of the molecule.
may be changed within some interval by shifting those phases.
Both above discussed schemes have been developed theoretically and require the knowledge of the PES of the molecular system to be controlled. They are characterized by a small number of control parameters which may be varied to optimize the probability of the formation of the desired product of the reaction. Indeed, both Tannor–Rice and Brumer–Shapiro schemes have been verified experimentally [HCP+00] (or see [RZ00] and references therein).
The theory of laser–pulse control has been finally put into a universal frame in suggesting the so-called Optimal Control (OC) theory by Rabitz and cowork-ers [PDR88, SWR88]. The OC theory is based on a certain functional which extremum has to be found. Once this extremum has been calculated the shape is known of the laser pulse which drives the system in a desired manner. The
Q
U(Q) |g>
|e>
|e >
w w w 3w
Figure 3.3: Control of dissociation into two different product states.
functional consists of the expectation value of the observable one wants to maximize (e.g. the population of a particular state) at a certain time, and a constrain which restricts the pulse energy to a finite value. It is believed that the above discussed direct control schemes can be reproduced by the OC theory when applying suitable constrains on the laser field. It will be demonstrated in this work for the case of Tannor-Rice scheme [TKR86].
Originally, the OC theory has been formulated for gas–phase systems which dynamics are governed by the time–dependent Schr¨odinger equation [PDR88, SWR88]. A formulation for mixed states could be already achieved in Ref.
[YGW+93]. The extension to reduced state dynamics of an open quantum system has been given in [BKT93], and recently in [YOR99] by extending the efficient iteration scheme of Refs. [ZBR98, ZR98].
Also the experiments on femtosecond laser–pulse control of molecular dy-namics became a subject of an active physico–chemical research [BG95,Zew97, Wil99]. Most of the work has been concentrated on the central idea of con-trolling chemical reactions resulting in a destruction or formation of a selected chemical bound. And indeed, a number of promising examples already exists even in the condensed phase [BDNG00]. While direct strategies according to
the above discussed schemes to achieve the control goal have been applied in an earlier state of this research (see [RZ00] and references therein) it was an exper-imental breakthrough to use highly flexible optical pulse shaping systems com-bined with self–learning algorithms as suggested in [JR92b]. Such techniques have been used for a broad variety of problems, among others for the control of fluorescence yields in dye molecules [BYW+97], to control yield of molecular photodissociation reactions [ABB+98, BBK+99, DFG+01], to control atomic two-photon transitions [MS98], to tailor atomic wavefunctions [WAB99], to excite selective molecular vibrations [WWB99] or to increase the efficiency of high-harmonic generation [BBz+00]. Beside the applications on the field of OC, the techniques developed to shape laser pulses and to use them for automatic optimal control experiments found also technological applications in telecomunications [WK98], in automated compression of ultrashort laser pulses [YMS97, BBS+97, EMB+98] or in multi-photon imagining techniques [BYS+99].
Although this type of approach found a widespread experimental applica-tion the use of self–learning algorithms in theory would remain on a preliminary level. This is because of the enormous amount of computational time necessary for carrying out the multitude of dynamic propagations. Consequently, it is much more appropriate to apply the OC theory whenever a control experiment has to be simulated.
The optimal laser fields obtained from the OC theory are usually charac-terized by a high complexity in contrast to the direct strategies as e.g. the pump-dump scheme. They may appear in the form of a complicated train of pulses with single features as short as few femtoseconds. It is almost cer-tainly a difficult task to create such pulses experimentally. Clearly, there is a lack of any restriction on the form of the optimal pulse in the standard formulation of OC theory we will present in the next section. This is an ad-vantage against the experimental approach with self-learning algorithms while searching for new control pathways but creates a significant obstacle for the application of the theoretical results in the experiments. Thus, the theory and experiment could not easily benefit from the advances of each other and their developments proceeded relatively independently in the past. This resulted in a significant incompatibility between both approaches. Namely, due to the different methodology the conditions put on the optimal search in the experi-ment and the theoretical considerations differ substantially. While in the initial stage the research on the OC of molecular dynamics has concentrated on devel-oping the methods of control and proving the controllability as such, recently efforts have been done to bring the experimental and theoretical approaches into closer cooperation [HMdVR01, HMdVR02, MM02].
It will be the aim of the following sections to formulate a general OC theory for the systems exhibiting dissipative dynamics and to study the possibility of their control by means of short laser pulses. Concerning the applicability of the theoretical results in OC experiments we will also discuss a standard experimental set-up for OC experiments and the ways of adapting the OC theory to the experimental conditions. First, in Chapters 3, 4 we concentrate on the prospects of the OC of electron transfer (ET) reactions also including the dissipative influence of the environment on the controlled dynamics. To this end we formulate the general OC for density matrix in Chapter 3 and apply it on diverse ET systems in Chapter 4. In Chapter 5 we introduce several generalizations of the OC and in Chapter5we discuss an example of the experimental set-up for OC experiments and its theoretical description and also formulates the OC theory in order to meet the requirements for the successful simulation of the OC experiments. In particular we address the problem of static disorder in the molecular ensemble and the optimization of the probe signal in the pump-probe experiment. The discussion from Chapter 5enables us to define certain notion of the complexity of the control task presented in section 5.2. Finally, to connect the OC theory for dissipative dynamics with Part II of this work we formulate the OC theory for non-Markovian dynamics in section 5.4.
3.1 Optimal Control Scheme for Dissipative Molecular Dynamics
The aim of this section is to derive an effective optimal control scheme valid for the optimization of the the control yields in open quantum systems. To this end we start with the most general type of OC theory, i.e. the formulation for a density operator [YOR99]. A reduction to mixed or pure state dynamics of closed system is straightforward and will be presented in the next section.
The time evolution of the density matrix has been discussed in detail in section (1.2). There, several types of dissipative dynamics have been studied with particular emphasis on memory effects. It was show there that these effects really induce different dynamics for the case of the system interaction with short laser pulses. On the other hand it became clear that in many cases their influence is not dramatic and some of them can be even reproduced by a certain corrections within the Markov theory. Also, to include memory effects proofed to be a very difficult task. Therefore, we will concentrate on the Markovian dynamics and thus omit the memory effects as well as the influence of the laser field on the dissipative part of the equation of motion. A possible
generalization of the OC theory for the case of the non-Markovian dynamics will be given in section 5.4.
Thus, the equation of motion for the reduced density matrix of the system to control reads
∂
∂tρ(t) =ˆ −iLmolρ(t)ˆ −iLF(t) ˆρ(t)− Dρ(t)ˆ . (3.3) Here, we comprised the whole dissipative influence of the thermodynamic reser-voir into a superoperator D. Its concrete form follows from the QME formu-lation in the Markov approximation (see section 1.3 and Appendix I. The system Hamiltonian is divided into a part describing the molecular system it-self Hmol and its interaction with the radiation field HF(t). The Lmol and LF are the Liouville superoperators corresponding to the commutators with Hmol
and HF(t), respectively. The time evolution of the reduced density operator can be expressed using the time evolution superoperator U(t, t0,E) as
ˆ
ρ(t) = U(t, t0;E) ˆρ(t0). (3.4) As we have already mentioned in the introduction to this chapter, the OC of molecular dynamics is usually formulated as the task to realize at a certain final time tf the expectation value
O(tf) = trS{Oˆρ(tˆ f)} (3.5) of the observable described by the (hermitian) quantum mechanical operator O. To getˆ O(tf) one applies a field pulseE(t) which should drive the system in the required manner (the optimal pulse). According to [YGW+93] the optimal pulse is defined as the extremum of the following functional
J(tf;E) =O(tf;E)− 1 2
tf
Z
t0
dt λ(t)E2(t), (3.6) where the second term on the right–hand side guarantees an upper limitation of the field intensity. (The penalty factor λ(t) has been taken time–dependent to avoid a sudden switch on and switch off of the control field [SdVR99].) A slightly different version of the functional has been suggested by Rabitz [ZBR98,ZR98,YOR99], who used a somewhat larger expression which ensures the use of the correct dynamic equations. In the present approach, however, the concrete dynamic equation to be used is already fixed by the demand how to determine O(tf), Eq. (3.5).
In order to determine the extremum one sets the functional derivative of J with respect to E equal to zero. One obtains (details of the derivation are given in the Appendix H)
E(t) = 1 λ(t)
δO(tf)
δE(t) = K(tf, t;E)
λ(t) . (3.7)
This expression has to be understood as a self–consistency relation for the optimal field. The actual value of the field at time t becomes proportional
This expression has to be understood as a self–consistency relation for the optimal field. The actual value of the field at time t becomes proportional